Random Walks, Capacity and Percolation on Trees

Abstract

A collection of several different probabilistic processes involving trees is shown to have an unexpected unity. This makes possible a fruitful interplay of these probabilistic processes. The processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well. Our work has five specific aims: First, an exact correspondence between random walks and percolation on trees is proved, extending and sharpening previous work of the author. This is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes. Second, we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation. This capacitary formulation extends and sharpens work of Fan on random interval coverings. Third, we show how this formulation also applies to generalize work of Evans on random labelling of trees. Fourth, the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a.s. In particular, we resolve a conjecture of Griffeath on the necessity of the Nash-Williams criterion. Fifth, for this last purpose, we establish several new basic results on branching processes in varying environments.